Lesson 4

Square Roots on the Number Line

Let’s approximate square roots.

Problem 1

  1. Find the exact length of each line segment.

    Linesegments AB and GH on grid. AB = square root 17, GH = square root 32.
  2. Estimate the length of each line segment to the nearest tenth of a unit. Explain your reasoning.

Problem 2

Plot each number on the \(x\)-axis: \(\sqrt{16},\text{ } \sqrt{35},\text{ } \sqrt{66}\). Consider using the grid to help.

quadrant 1, x axis, 0 to 10, by 1's. y axis, 0 to 8, by 1's. 


Problem 3

Use the fact that \(\sqrt{7}\) is a solution to the equation \(x^2 = 7\) to find a decimal approximation of \(\sqrt{7}\) whose square is between 6.9 and 7.1.


Problem 4

  1. Explain how you know that \(\sqrt{37}\) is a little more than 6.

  2. Explain how you know that \(\sqrt{95}\) is a little less than 10.

  3. Explain how you know that \(\sqrt{30}\) is between 5 and 6.

Problem 5

Plot each number on the number line: \(\displaystyle 6, \sqrt{83}, \sqrt{40}, \sqrt{64}, 7.5\)

A number line with 6 evenly spaced tick marks and the integers 5 through 10 are indicated.

Problem 6

The equation \(x^2=25\) has two solutions. This is because both \(5 \boldcdot 5 = 25\), and also \(\text-5 \boldcdot \text-5 = 25\). So, 5 is a solution, and also -5 is a solution.

Select all the equations that have a solution of -4:













Problem 7

Find all the solutions to each equation.

  1. \(x^2=81\)
  2. \(x^2=100\)
  3. \(\sqrt{x}=12\)

Problem 8

The points \((12, 23)\) and \((14, 45)\) lie on a line. What is the slope of the line?

(From Unit 5, Lesson 4.)